5. Euler, Euler, Master of Us All

This chapter delves into the calculus of variations, a foundational method in dynamic optimization. It begins by distinguishing static optimization, which deals with single-variable calculus, from dynamic optimization, which involves variations in entire functions over time. The text introduces the concept of functionals, which take entire functions as inputs and return single values, and explains how these are used to find optimal paths in dynamic problems. The chapter highlights the work of Leonard Euler, who perfected the calculus of variations, and discusses its historical significance. It then explores the Euler equation, the first-order condition for optimality in the calculus of variations, and provides intuitive explanations and examples to illustrate its application. The chapter also covers the expanded form of the Euler equation and its solution, as well as transversality conditions, which are crucial for problems with variable endpoints. Through numerical examples and economic applications, the text demonstrates how the calculus of variations can be used to solve real-world problems, such as optimizing consumption paths and investment strategies. The chapter concludes by discussing the limitations of the calculus of variations and introduces other branches of dynamic optimization, such as optimal control theory and dynamic programming.